Hello,
When calculating the angle of projection or the angle a particle strikes the ground for a vector function, do you use tanθ=dy/dx (when t=0 for projection or t=time particle hits ground)? Why don't you use tanθ=y/x, and is there any time where you do use this? Is this different from calculating the angle of elevation at a given time?
Thanks, I'm just for some reason really confused about this.
I think you may be thinking too much of formulas that you regurgitate, instead of thinking about the physical system. Golden rule for maths - if you can graph it, you should graph it. So, consider the picture I've attached to this post.
This picture shows the parabolic motion for whatever is moving along your typical projectile motion. Maybe the motion isn't parabolic - but let's pretend it is, just for this picture. If you want to figure out the angle a particle is projected at, or the angle it'll land the ground at, then you need to draw a triangle, and use SOH-CAH-TOA to figure out the angle.
So, can we use the displacement curve? Well, no - because we're interested in the angle at a point, and this triangle has straight lines that only touch the curve once. In fact, that one time is tangent to the curve - so maybe we can use the derivative? In fact, that triangle's two side lengths are going to be dy/dt and dx/dt, because that's the derivative - and so, the the triangle as a whole should have:
Which is the equation we wanted! Important to note - we didn't use the fact that our motion was "parabolic" in any way, so this "proof" will still work, REGARDLESS of the type of motion. However, by pretending it was and drawing it out, we were able to come to a better understanding of what was going on, and come up with a formula to solve for the setting we were in. Do you think, being able to see this diagram, you can think of any time we might use a formula tan(theta)=y/x? If you can't think of any time we might see this, why do you think we might not? This is just a question to test your understanding, so just give it a go, you have nothing to lose if you get it wrong